the sturm -liouville problem - university of groningen...abstract this thesis treats the...

faculty of science and engineering

The Sturm-Liouville Problem

Master Project Mathematics (Science, Business & Policy track)

July 2017

Student: D.E. Koning

First supervisor: Dr. A.E. Sterk

Second supervisor: Prof. dr. H. Waalkens

Abstract

This thesis treats the Sturm-Liouville problem, a typical case of somedifferential equation subject to certain boundary conditions. After a shortintroduction this problem is explored using the separation of variablesmethod to solve a differential equation associated with the Sturm-Liouvilletheory. Then some basic properties of vector spaces and inner productspaces will be given and the L2 space will be discussed, which is oneof the most common examples of a Hilbert space. This enables us todescribe the convergence, completeness and orthogonality of functions inthe L2 space. Using this, differential operators which are self-adjoint willbe explored since this concept applies to the operator associated with theSturm-Liouville problem. Thereafter the Sturm-Liouville problem itselfshall be discussed and some spectral properties concerning the eigenvaluesand eigenfunctions will be proven. Finally the theory discussed thus farwill be illustrated by working out an example and some applications ofthis subject will be given.

CONTENTS

Contents

1 Introduction 1

2 The Sturm-Liouville Problem 22.1 Ordinary differential equations . . . . . . . . . . . . . . . . . . . 22.2 The Sturm-Liouville problem . . . . . . . . . . . . . . . . . . . . 32.3 The separation of variables method . . . . . . . . . . . . . . . . . 4

2.3.1 Solution of a Sturm-Liouville equation . . . . . . . . . . . 42.3.2 The heat equation . . . . . . . . . . . . . . . . . . . . . . 5

3 Vector Spaces 83.1 The inner product space . . . . . . . . . . . . . . . . . . . . . . . 83.2 The L2 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Convergence in L2 . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Orthogonal functions . . . . . . . . . . . . . . . . . . . . . 13

4 Self-Adjoint Differential Operators 18

5 The Spectral Theory 225.1 Existence of the eigenvalues and eigenfunctions . . . . . . . . . . 235.2 Completeness of the eigenfunctions . . . . . . . . . . . . . . . . . 30

6 Application of the Sturm-Liouville Theory 346.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 An example: the second derivative . . . . . . . . . . . . . . . . . 356.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 37

7 Acknowledgements 38

8 References 39

ii

1 INTRODUCTION

1 Introduction

Probably everyone has heard or played an instrument, for example a guitar. Thesound of a guitar comes from plucking a string. The vibration of the pluckedstring changes the initial position and initial velocity. The motion of the vi-brating string can be described by a wave equation, which is a typical problemthat can be solved by the Sturm-Liouville theory. Exploring some of its theproperties gives us insight in the harmonics of the instrument, which clarifiesthe pleasing sound of a guitar.Not only the sound of a guitar, but many other important physical processesand mechanical systems from classical physics and quantum physics can be de-scribed by means of a Sturm-Liouville equation. This Sturm-Liouville theorydeals with linear second-order differential equations subject to particular bound-ary conditions. Often these differential equations describe the oscillation of e.g.pendula, vibrations, resonances or waves.The Sturm-Liouville problem is named after Jacques Charles Francois Sturm(1803–1855) and Joseph Liouville (1809–1882). Sturm described how a partialdifferential equation corresponding to a Sturm-Liouville problem can be solvedby the separation of variables method. He discussed the heat conduction in aninhomogeneous thin bar as an example to illustrate this. However, before himpeople were mostly interested in the specific solution itself. In contrast, Sturmand Liouville came up with a theory which focused on investigating the prop-erties of the solution of the differential equation, such as the eigenfunctions. Inview of the guitar problem the eigenfunctions correspond to the resonant fre-quencies of vibration. For a more detailed and very interesting depiction of theorigin and the history of the Sturm-Liouville problem the reader is referred topages 423-475 of [7].The simplest example of a Sturm-Liouville operator is the constant-coefficientsecond-derivative operator. This example will be elaborated at the end of thisthesis and we shall see the eigenfunctions are the trigonometric functions. Otherfunctions associated to Sturm-Liouville operators are the well-known Airy func-tion and Bessel function.

First in chapter 2 a brief overview of ordinary differential equations and the gen-eral Sturm-Liouville theory will be given. The separation of variables methodwill be utilized to give a solution to this problem and to solve a particularcase, namely the heat equation. Then in chapter 3 some basic properties ofvector spaces and inner product spaces will be given. This enables us to dis-cuss the L2 space, which is a particular case of a Hilbert space. Finally theconvergence, completeness and orthogonality of functions in the L2 space willbe described in this section. Thereafter this will be used to explore the self-adjointness property of differential operators in chapter 4. We will show theSturm-Liouville operator treated in this thesis is also self-adjoint. In chapter 5the Sturm-Liouville problem itself will be discussed together with its spectralproperties. Several statements concerning the eigenvalues and eigenfunctionswill be proven. Finally in section 6 some applications of the Sturm-Liouvilletheory will be given, whereafter a Sturm-Liouville problem will be illustrated.The second-derivative operator will be studied in detail as a concrete example,followed by some concluding remarks.

1

2 THE STURM-LIOUVILLE PROBLEM

2 The Sturm-Liouville Problem

In this section we start with some basic concepts of ordinary differential equa-tions. Then we will give the general theory behind the Sturm-Liouville prob-lem and afterwards we shall shortly give a review of the separation of variablesmethod for solving partial differential equations. However, we assume the readeris somewhat familiar with this method and partial differential equations in gen-eral. At the end of this section this method will be utilized to solve a differentialequation associated with the Sturm-Liouville problem.

2.1 Ordinary differential equations

Let’s first look at ordinary differential equations in general. Consider an ordi-nary differential equation of second order on the real interval I given by thefollowing equation

α0(x)u′′ + α1(x)u′ + α2(x)u = f(x), (1)

where α0, α1, α2 and f are complex functions on I. If the function f equals zero,equation (1) is homogeneous. Otherwise it is considered as inhomogeneous, i.e.f 6= 0. If we define the following second-order differential operator

L = α0(x)d2

dx2+ α1(x)

d

dx+ α2(x),

we are able to rewrite equation (1) in the following form

Lu = f.

Consider any two functions f, g for which the first and second derivatives arecontinuous on the interval I, i.e. f, g ∈ C2(I). If they are solutions of differentialequation (1), then for constants c1, c2 ∈ C we have

L(c1f + c2g) = c1Lf + c2Lg,

which implies the operator L is linear. Therefore we call (1) a linear differen-tial equation. If f and g are solutions to the linear homogeneous differentialequation, i.e.

Lf = 0 and Lg = 0,

then any linear combination of solutions is also a solution. This implies

L(c1f + c2g) = c1Lf + c2Lg = 0.

If the function α0 is nowhere zero on the interval I we can divide the ordinarydifferential equation (1) by α0 which gives

u′′ + q(x)u′ + r(x)u = g(x), (2)

where q = α1

α0, r = α2

α0and g = f

α0. Obviously, the ordinary differential equations

(1) and (2) have the same set of solutions and in this case equation (1) iscalled regular. The equation is said to be singular if there exists some point

2


z ∈ I such that α0(z) = 0 and in this case z is called the singular point ofthe ordinary differential equation. Since we assume the reader is familiar withlinear second-order differential equations we neither discuss initial conditionsand boundary conditions, nor the properties of the solutions of equation (2).Later on, especially from section 4 about self-adjoint operators onwards, we willmake use of the statements made in this section.

2.2 The Sturm-Liouville problem

The Sturm-Liouville problem is a differential equation over the finite interval[a, b] which is of the following form

− d

dx

[p(x)

du(x)

dx

]+ r(x)u(x) = λw(x)u(x). (3)

The Sturm-Liouville problem is called regular if the functions p(x) and w(x) areboth strict positive, p(x), p′(x), r(x) and w(x) are continuous over the interval[a, b] and moreover it has the following non-trivial boundary conditions

α1u(a) + α2u′(a) = 0 (α2

1 + α22 > 0),

β1u(b) + β2u′(b) = 0 (β2

1 + β22 > 0).

(4)

Of course the trivial solution u(x) = 0 is always a solution. However, thenon-trivial solutions of the differential equation given in (3) which satisfy theboundary conditions in (4) only exist for specific values of λ. These values arecalled the eigenvalues of the boundary value problem and we denote them byλn. The corresponding eigenfunctions are denoted by un. Before solving theproblem using the separation of variables method we will first give some mainresults of the Sturm-Liouville theory which among others can be found on pages147-148 of [8] or pages 72-73 of [13].

1. The eigenvalues can be ordered in the following way

λ1 < λ2 < λ3 < · · · < λn < · · · ,

where limn→∞ λn →∞. These eigenvalues show asymptotic behaviour ascan be seen in the example given in section 6. However, this will not beproven in this thesis but can be found on page 275 of [12].

2. The eigenfunction un corresponding to the eigenvalue λn is unique up toa constant normalization factor and it has exactly n− 1 zeros in the openinterval (a, b).

3. After normalizing un(x), the eigenfunctions form an orthonormal basis,i.e.

〈un, um〉 =

∫ b

a

un(x)um(x)w(x) dx = δmn.

This is an orthonormal basis for the weighting function w(x) over theinterval [a, b] in the Hilbert space L2([a, b], w(x)dx). Later on in this thesiswe will clarify the notion of Hilbert spaces and the L2 space.

3


2.3 The separation of variables method

We will now give a short review of the separation of variables method. Thismethod is used for solving linear partial differential equations with boundaryand initial conditions. Examples are the heat equation, wave equation, Laplaceequation and Helmholtz equation, which we assume the reader are familiar with.This method will be illustrated in section 2.3.1 by solving a Sturm-Liouvillepartial differential equation. Then in section 2.3.2 we will consider a specialcase where the separation of variable method will be utilized to give a solutionto the homogeneous heat equation.

2.3.1 Solution of a Sturm-Liouville equation

Let’s consider a partial differential equation of the following form with corre-sponding boundary and initial conditions

f(x)∂2u

∂x2+ g(x)

∂u

∂x+ h(x)u =

∂u

∂t+ k(t)u,

u(a, t) = u(b, t) = 0,

u(x, 0) = s(x).

(5)

The separation of variables method is based on u being writable as a product oftwo functions X and T where the variables are separated, as the name alreadysuggested. So a solution would be of the following form

u(x, t) = X(x)T (t).

Using this expression for u(x, t) and taking the appropriate derivatives enablesus to write the equation given in (5) in the following separated form

LX(x)

X(x)=MT (t)

T (t), (6)

where we used the following substitutions

L = f(x)d2

dx2+ g(x)

d

dx+ h(x),

M =d

dt+ k(t).

(7)

We see L and X(x) only depend on the variable x and M and T (t) only on t.Since the right hand side only depends on x and the left hand side only on t, weknow both sides of (6) have to be equal to some constant which we will denoteby λ. Therefore we can write our differential equation in the following form

LX(x) = λX(x),

MT (t) = λT (t).(8)

Using the boundary conditions given in (5) we obtain

X(a) = X(b) = 0.

4


The Sturm-Liouville theory enables us to solve the first formula of (8) and fornow we assume this solution is known, i.e. we already have the eigenfunctionsXn and eigenvalues λn. However, we are already able to solve the second formulaof (8). Again assuming the eigenvalues are known and substituting the formulafor M given in (7) we obtain

d

dtTn(t) = (λn − k(t))Tn(t).

This leads to the following solution for Tn

Tn(t) = ane(λnt−

∫ t0k(τ)dτ).

Using this formula and substituting it in the equation for u(x, t) results in

u(x, t) =∑n

anXn(x)e(λnt−∫ t0k(τ)dτ).

The coefficient an can be determined using the Hilbert space L2 with its corre-sponding inner product, which will be discussed later on in this thesis.

2.3.2 The heat equation

Let’s examine the heat equation, which is a special case of a Sturm-Liouvilleproblem. In the one-dimensional form it is given by the following formula

∂u

∂t− α∂

2u

∂x2= 0, (9)

and we consider homogeneous boundary conditions, i.e.

u(0, t) = 0 = u(L, t). (10)

Now we will separate the variables, which implies we consider a solution of x ofthe following form

u(x, t) = X(x)T (t), (11)

as we have seen before. Taking the first and second partial derivative of thefunction u(x, t) given in (11) with respect to t and x, making use of the productrule, and substituting this in the heat equation (9) we obtain the following result

T ′(t)

αT (t)=X ′′(x)

X(x).

Again this implies both sides of the equation above are equal to some constant−λ, leading to the following expressions

T ′(t) = −λαT (t),

X ′′(x) = −λX(x).

For non-positive λ there are no solutions for X(x), as we shall see when consid-ering the following two cases. Let us first assume λ < 0. Then a solution for Xis given by

X(x) = Be√−λx + Ce−

√−λx,

5


where B,C ∈ R. Then, using the homogeneous boundary conditions given in(10) we obtain

X(0) = 0 = X(L),

which implies B = 0 = C. Therefore u = 0.Assuming λ equals 0 we have the following solution for X

X(x) = Bx+ C,

where B,C ∈ R. Using the boundary conditions again we obtain u = 0.This implies λ has to be positive. This gives the following solutions for T andX

T (t) = Ae−λαt,

X(x) = B sin(√λx) + C cos(

√λx),

where A,B,C ∈ R. Then from the boundary conditions, i.e. X(0) = 0 = X(L)we obtain

C = 0 and√λ = n

π

L,

for some n ∈ N. So this is the solution to the heat equation if we can writeu(x, t) as a product where the variables x and t are separated as in (11).Summing over the solutions satisfying the heat equation with the boundaryconditions given in (10) yields

u(x, t) =

∞∑n=1

Dn sinnπx

Lexp

(−n

2π2αt

L2

). (12)

This is also a solution to the heat equation satisfying the boundary conditionsand therefore we can regard it as the final solution. Using an initial conditionenables us to determine Dn. If we are given the following initial condition

u(x, 0) = f(x),

we obtain the following solution for f(x) by substituting t = 0 in equation (12)

f(x) =

∞∑n=1

Dn sinnπx

L.

If we multiply both sides of the last equation by sin nπxL and integrate them

from 0 to L we obtain the following result for Dn

f(x) sinnπx

L=

∞∑n=1

Dn

(sin

nπx

L

)2∫ L

0

f(x) sinnπx

Ldx =

∫ L

0

∞∑n=1

Dn

(sin

nπx

L

)2dx

∫ L

0

f(x) sinnπx

Ldx =

L

2Dn

Dn =2

L

∫ L

0

f(x) sinnπx

Ldx.

6


Now the Sturm-Liouville theory plays an important role since it assures us theeigenfunctions, which in this case are given by

{sin nπx

L

}∞n=1

, are orthogonaland complete. The orthogonality and completeness of the eigenfunctions arenecessary conditions for this method and will be discussed later on in this thesis.

7

3 VECTOR SPACES

3 Vector Spaces

In this section some basic concepts shall be given which are necessary to betterunderstand the rest of the thesis. Definitions and theorems from metric spaceswill be omitted, since they should be known by the reader. For a detailedtreatment of this subject, the reader is referred to [11].We shall consider a linear vector space X, or just called a vector space. Thisis a set over K whose elements are called vectors and we have two operations,namely addition and scalar multiplication. K here stands for the field of realnumbers R or the field of complex numbers C. We assume the reader is familiarwith the basic properties of linear algebra, but for an accessible course on thistopic we recommend [6].First in section 3.1 some basic properties of an inner product space will begiven, which is a special type of a vector space. This enables us to discuss theL2 space in chapter 3.2, which is a particular case of a Hilbert space. At the endof this section we will describe the convergence, completeness and orthogonalityof functions in the L2 space.

3.1 The inner product space

The vector space X is called an inner product space if for every pair of elementsx, y ∈ X we have a corresponding inner product, denoted as 〈x, y〉 ∈ K. Thiscan be seen as a function from X × X → K such that for x, y ∈ X we have(x, y) 7→ 〈x, y〉 ∈ K. The inner product has to satisfy the following properties

1. The inner product of two elements equals the complex conjugate of thereversed inner product, i.e

〈x, y〉 = 〈y, x〉.

2. The inner product is linear, so for x, y, z ∈ X and α ∈ K this implies

〈x+ y, z〉 = 〈x, z〉+ 〈y, z〉 and

〈αx, y〉 = α〈x, y〉.

3. The inner product of an element with itself is positive definite, so for allx ∈ X we have

〈x, x〉 ≥ 0 and

〈x, x〉 = 0 if and only if x = 0.

An inner product space satisfies the following condition, which is also known asthe Cauchy-Schwarz Inequality.

Theorem 3.1. If X is an inner product space, then for all x and y in X wehave

|〈x, y〉|2 ≤ 〈x, x〉〈y, y〉.Proof. Let α be any complex number. Then, using the properties an innerproduct has to satisfy on an inner product space, we get the following inequality

0 ≤ 〈x− αy, x− αy〉= 〈x, x〉+ 〈x,−αy〉+ 〈−αy, x〉+ 〈−αy,−αy〉

= 〈x, x〉 − α〈x, y〉 − α〈x, y〉+ αα〈y, y〉,(13)

8

3 VECTOR SPACES

where α denotes the complex conjugate of α.

Let us first consider the case where 〈y, y〉 6= 0. Taking α = 〈x,y〉〈y,y〉 and substituting

into the inequality given in (13) gives

0 ≤ 〈x, x〉 − 〈x, y〉〈y, y〉

〈x, y〉 − 〈x, y〉〈y, y〉

〈x, y〉+〈x, y〉〈y, y〉

〈x, y〉〈y, y〉

〈y, y〉

= 〈x, x〉 − |〈x, y〉|2

〈y, y〉− |〈x, y〉|

2

〈y, y〉+|〈x, y〉|2

〈y, y〉

= 〈x, x〉 − |〈x, y〉|2

〈y, y〉,

which is exactly the inequality we wanted to prove.Now we consider the case where 〈y, y〉 = 0. This is actually trivial since itimplies y = 0 and therefore the inequality always holds. This makes the proofcomplete.

The vector space X is a normed space if for every element of X there exists anorm, which is a non-negative real number. If we have a vector x ∈ X we willdenote the norm by ‖x‖ and it satisfies the following properties

1. ‖x+ y‖ ≤ ‖x‖+ ‖y‖

2. ‖αx‖ = |α|‖x‖

3. ‖x‖ > 0 whenever x 6= 0,

for all x, y ∈ X and scalars α.Now let’s define the following norm for x ∈ X

‖x‖ = 〈x, x〉 12 , (14)

where x is called normalized if we have ‖x‖ = 1. Using this norm the Cauchy-Schwarz inequality given in theorem 3.1 takes on the following form

|〈x, y〉| ≤ ‖x‖‖y‖. (15)

This defines a norm on an inner product space X if we prove the followingtheorem.

Theorem 3.2. If X is an inner product space, then for all x and y in X wehave

‖x+ y‖ ≤ ‖x‖+ ‖y‖.

Proving this theorem implies every pair of elements of X satisfies the threeproperties above. Therefore this defines a norm on an inner product space X.

Proof. Using the norm given in (14) we obtain

‖x+ y‖2 = 〈x+ y, x+ y〉= ‖x‖2 + 〈x, y〉+ 〈y, x〉+ ‖y‖2

= ‖x‖2 + 2 Re〈x, y〉+ ‖y‖2

≤ ‖x‖2 + 2|〈x, y〉|+ ‖y‖2

≤ ‖x‖2 + 2‖x‖‖y‖+ ‖y‖2 using (15)

= (‖x‖+ ‖y‖)2

9

3 VECTOR SPACES

Therefore, taking square roots of both sides, we obtain the following inequality

‖x+ y‖ ≤ ‖x‖+ ‖y‖,

which proves the theorem.

Theorem 3.2 also leads to the well-known triangle inequality. defining the dis-tance between two elements x and y as ‖x − y‖, we obtain for three elementsx, y, z ∈ X the following inequality

‖x− y‖ = ‖x− z + z − y‖ ≤ ‖x− z‖+ ‖z − y‖. (16)

Now we are able to explore the idea of orthogonal sets.

Definition 3.3. If we have 〈x, y〉 = 0 for x, y ∈ X, we call x orthogonal to y,denoted as x ⊥ y. A subset S ⊆ X is called an orthogonal set if every pair ofelements in this subset is orthogonal.

We can see definition 3.3 for the n-dimensional real space also intuitively. If wehave two vectors x, y ∈ Rn we can define the angle θ between these two vectorsusing the Cauchy-Schwarz inequality given in (15) in the following way

〈x, y〉 = ‖x‖‖y‖ cos θ.

Notice we used the term cos θ to make an equality of the inequality. This angleθ is defined uniquely and if the vectors x and y are both nonzero we must have

〈x, y〉 = 0 if and only if cos θ = 0,

which is exactly the condition for the two vectors x and y to be orthogonal inRn. Using the definition of an orthogonal set we are able to give the concept ofan orthonormal set.

Definition 3.4. An orthogonal subset S of an inner product space X is calledorthonormal if every element of this subset is normalized, i.e. the norm of everyelement x ∈ S equals 1 (or symbolically ‖x‖ = 1).

3.2 The L2 Space

Now we are able to move on to the L2 space, which is named after the Frenchmathematician Henri Lebesgue (1875 – 1941). First let us define the vectorspace denoted by C ([a, b]). This vector space consists of complex continuousfunctions on the real interval [a, b]. For two functions f, g ∈ C ([a, b]) we definethe following inner product

〈f, g〉 =

∫ b

a

f(x)g(x) dx. (17)

Using the norm given in (14) we obtain

‖f‖ = 〈f, f〉 12 =

√∫ b

a

f(x)f(x) dx =

√∫ b

a

|f(x)|2 dx. (18)

We can now show that the Cauchy-Schwarz inequality given in theorem 3.1 andthe triangle inequality given in equation (16) both still hold in the vector spaceC([a, b]). This boils down to proving (17) is an inner product, so we have tocheck the three properties an inner product has to satisfy.

10

3 VECTOR SPACES

Proof.

1. The inner product of two elements equals the complex conjugate of thereversed inner product, i.e.

〈f, g〉 =

∫ b

a

f(x)g(x) dx =

∫ b

a

g(x)f(x) dx = 〈g, f〉.

2. The inner product is linear, so for f, g, h ∈ C ([a, b]) and α ∈ K we have

〈f + g, h〉 =

∫ b

a

(f(x) + g(x))h(x) dx

=

∫ b

a

f(x)h(x) dx+

∫ b

a

g(x)h(x) dx = 〈f, h〉+ 〈g, h〉,

and

〈αf, g〉 =

∫ b

a

αf(x)g(x) dx = α

∫ b

a

f(x)g(x) dx = α〈f, g〉.

3. The inner product is positive definite, so for f, g ∈ C ([a, b]) we have

〈f, f〉 =

∫ b

a

f(x)f(x) dx =

∫ b

a

|f(x)|2 dx ≥ 0,

Furthermore since g is continuous we must have 〈g, g〉 ≥ 0, so in case thisis an equality this must imply g ≡ 0.

Therefore, the inner product defined by (17) satisfies both the Cauchy-Schwarzinequality and the triangle inequality. Thus we have

|〈f, g〉| ≤ ‖f‖‖g‖ and ‖f + g‖ ≤ ‖f‖+ ‖g‖. (19)

We will now use L2(a, b) to denote the set of functions f : [a, b]→ C for whichthe following hold ∫ b

a

|f(x)|2 dx <∞. (20)

However we have to note that these functions have to be Lebesgue measurable.This Lebesgue measure is a natural extension of the classical notions of lengthsand areas to more general sets. It is a standard way of measuring a set by cov-ering it with intervals and then taking the sum of the lengths of these intervalsas an approximation of the measure of the set. This subject will not be furtherdiscussed, but a brief review can be found on pages 110-112 of [4] or for a moreextensive description the reader is referred to sections 16-20 of [3].

Now we will define the same inner product as in (17) and norm as in (18) onL2(a, b). Applying the triangle inequality on the following norm yields

‖αf + βg‖ ≤ ‖αf‖+ ‖βg‖ = |α|‖f‖+ |β|‖g‖,

11

3 VECTOR SPACES

for all f, g ∈ L2(a, b) and α, β ∈ C. This implies αf + βg ∈ L2(a, b), followingfrom the fact that the norm of f and g is less than infinity since f, g ∈ L2(a, b).Therefore it is a vector space and with this inner product it becomes an innerproduct space. Actually C([a, b]) is a subspace of L2(a, b) and from now on wewill use the latter space since the integrals are interpreted as Lebesgue integrals.This is necessary since from the Cauchy-Schwarz inequality it follows that theinner products of f and g are well defined if ‖f‖ and ‖g‖ exist. This implies|f |2 and |g|2 have to be integrable and this is exactly the case for L2(a, b) sinceit consists of those functions f such that |f |2 is integrable on [a, b]. Again moreinformation about Lebesgue integrals can be found on pages 108-110 of [4] orpages 322-323 of [3].

Two functions f, g ∈ L2(a, b) are called equal almost everywhere if ‖f − g‖ = 0.This term comes from the fact f and g do not have to be equal pointwise onthis interval, i.e. f(x)− g(x) = 0 at every x ∈ [a, b]. This is due to the fact thatin this space ‖f‖ = 0 does not always imply f(x) = 0 at every point x ∈ [a, b].Therefore we can regard the space L2 as been made up out of equivalence classesof functions which are equal almost everywhere.

3.2.1 Convergence in L2

Before describing convergence in the space L2 we will first give the definitionsof pointwise and uniform convergence of a sequence of functions.

Definition 3.5. i) A sequence of functions fn : I → F converges pointwiseto the function f : I → F if for every x ∈ I we have limn→∞ fn(x) = f(x).

i)) This sequence of functions is said to be uniformly convergent if for everyε > 0 there exists a natural number N such that

n ≥ N implies |fn(x)− f(x)| < ε for all x ∈ I.

We will denote pointwise convergence by fn → f and uniform convergence byfn

u→ f . One nice consequence is that uniform convergence implies pointwiseconvergence (but not necessarily the other way around).

Now we have given the definitions of pointwise and uniform convergence of asequence of functions, we will mostly be looking at convergence in the space L2

in this section.

Definition 3.6. If we have a sequence of functions (fn) in L2(a, b), it convergesin L2 if there exists a function f ∈ L2(a, b) such that

limn→∞

‖fn − f‖ = 0.

In other words, for every ε > 0 there exists a natural number N such that

n ≥ N implies ‖fn − f‖ < ε.

We will denote convergence in L2 by fnL2

→ f , where f is said to be the limit inL2 of the sequence (fn).

12

3 VECTOR SPACES

Pointwise convergence does not imply convergence in L2 and convergence in L2

does also not imply pointwise convergence. This is due to the limit in L2 isan equivalence class of functions, i.e. they are equal in L2 but not pointwise.However, if a sequence of functions converges pointwise as well as in L2, thenthese limits are equal.

Uniform convergence on the other hand does imply convergence in L2 underthe following conditions. The sequence (fn) and its limit f must lie in L2(I)and this interval I should be bounded. The proof will not be given since it usessome basic properties of uniform convergence which have not been discussed inthis thesis.

In the preliminaries we assumed the reader is familiar with metric spaces andtherefore knows concepts such as completeness and the Cauchy sequence. Nowwe shall give these definitions in terms of the L2 space.

Definition 3.7. A sequence in L2 is called a Cauchy sequence if for each ε > 0there exists a natural number N such that ‖fn − fm‖ < ε whenever n,m ≥ N .

A sequence (fn) which converges in L2 is a Cauchy sequence. Namely, usingthe triangle inequality we have

‖fn − fm‖ ≤ ‖fn − f‖+ ‖f − fm‖.

If we take m and n large enough, the right-hand side of the last inequality can

be made as small as desired since we have fnL2

→ f . Therefore the left handside can be made arbitrarily small, which implies this convergent sequence is aCauchy sequence.

Furthermore the space L2 is complete. This is because it is one of the mostcommon examples of a Hilbert space, which is a complete inner product spaceunder its norm induced by the inner product. Completeness of the L2 spaceactually means the following.

Theorem 3.8. For every Cauchy sequence (fn) in L2 there exists a function

f ∈ L2 such that fnL2

→ f .

The proof can be found on pages 329-330 of [10]. Moreover, the set of continuousfunctions is dense in this space, i.e. C([a, b]) is dense in L2(a, b). This densityproperty is given by the following theorem and can also be found in [10] on page326.

Theorem 3.9. For any f ∈ L2(a, b) and any ε > 0 there exist a continuousfunction g on [a, b] such that ‖f − g‖ < ε.

In section 5.2 this theorem will be used to prove the eigenfunctions of a Sturm-Liouville problem are complete, but for now we will first have a look at theorthogonality property in the L2 space.

3.2.2 Orthogonal functions

Let’s say we want to write a function f ∈ L2 in terms of other functions inthe complex space L2. So suppose we have an orthogonal set of functions in

13

3 VECTOR SPACES

L2, i.e. {ϕ1, ϕ2, ϕ3, . . . }. Note that this set could be finite or infinite. If thefunction f is a finite linear combination of elements in this set {ϕk}, then f canbe represented as follows

f =

n∑i=1

αiϕi with αi ∈ C. (21)

Since we are dealing with an orthogonal set the inner product of every elementwith another element equals zero as we have seen in definition 3.3, i.e.

〈ϕi, ϕj〉 = 0, whenever i 6= j.

Therefore, taking the inner product of ϕk with f defined as in (21) yields

〈f, ϕk〉 = αk ‖ϕk‖2. (22)

Dividing both sides by ‖ϕk‖2 we obtain an expression for the coefficient αk

αk =〈f, ϕk〉‖ϕk‖2

. (23)

Substituting this into equation (21) we see the function f can be represented inL2 as follows

f =

n∑k=1

〈f, ϕk〉‖ϕk‖2

ϕk. (24)

So we see the coefficients αk are determined by what we call the projections of fon ϕk. We can also obtain an orthonormal set by dividing each ϕi by its norm,

i.e.{ψk = ϕk

‖ϕk‖

}. This defines an orthonormal set since every element ψk is

normalized. In other words ‖ψk‖ = 1 as we saw in definition 3.4. Thereforeequation (22) becomes 〈f, ψk〉 = αk‖ψk‖2 = αk and it enables us to rewrite ourfunction f as in (24) as follows

f =

n∑k=1

〈f, ψk〉ψk,

where the coefficients are the projections of f on ψk.

On the other hand, if f is not a linear combination of elements of {ϕk} wewould like to determine its best approximation in L2, again using a finite linearcombination of the elements of {ϕk}. This comes down to determining thecoefficients αk for which the following norm is minimized∥∥∥∥∥f −

n∑k=1

αkϕk

∥∥∥∥∥ . (25)

If we apply the definition of the norm to the square of the norm given above we

14

3 VECTOR SPACES

get ∥∥∥∥∥f −n∑k=1

αkϕk

∥∥∥∥∥2

=

⟨f −

n∑k=1

αkϕk, f −n∑k=1

αkϕk

⟩

= ‖f‖2 − 2

n∑k=1

Reαk〈f, ϕk〉+

n∑k=1

|αk|2‖ϕk‖2

= ‖f‖2 −n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2

+

n∑k=1

‖ϕk‖2[|αk|2 − 2Reαk


+|〈f, ϕk〉|2

‖ϕk‖4

]

= ‖f‖2 −n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2+

n∑k=1

‖ϕk‖2∣∣∣∣αk − 〈f, ϕk〉‖ϕk‖2

∣∣∣∣2 .We only have to deal with the last term of this equation since we want todetermine the coefficients αk which minimize the norm given in (25). Thislast term on the right hand side above is always greater or equal to zero so tominimize ‖f −

∑k=1 αkϕk‖2 (and therefore also its square root) we want to

choose αk such that the last term equals zero, i.e.

αk =〈f, ϕk〉‖ϕk‖2

.

This makes the last term∑nk=1 ‖ϕk‖2

∣∣∣αk − 〈f,ϕk〉‖ϕk‖2

∣∣∣2 equal to zero and therefore

we obtained the minimum of the norm given in (25). Substituting this αk inour formula for the squared norm yields∥∥∥∥∥f −

n∑k=1


ϕk

∥∥∥∥∥2

= ‖f‖2 − 2

n∑k=1

Re〈f, ϕk〉〈f, ϕk〉‖ϕk‖2

+

n∑k=1

|〈f, ϕk〉|2

‖ϕk‖4‖ϕk‖2

= ‖f‖2 − 2

n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2+

n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2

= ‖f‖2 −n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2.

Since the left hand side has to be greater or equal to zero we can rewrite theright hand side to obtain the following result

n∑k=1

|〈f, ϕk〉|2

‖ϕk‖2≤ ‖f‖2.

Since this holds for any n we can take the limit as n→∞ which gives

∞∑k=1

|〈f, ϕk〉|2

‖ϕk‖2≤ ‖f‖2. (26)

15

3 VECTOR SPACES

This last relation is called Bessel’s inequality and it holds for any orthogonalset {ϕk : k ∈ N} and any f ∈ L2.This inequality becomes an equality if and only if∥∥∥∥∥f −

∞∑k=1


ϕk

∥∥∥∥∥ = 0.

This actually means

f =

∞∑k=1


ϕk in L2.

Therefore we see the sum∑∞k=1 αkϕk with αk = 〈f,ϕk〉

‖ϕk‖2 is a representation of

the function f in L2.We will now use these representations of f to describe the completeness of anorthogonal set making use of theorem 3.8. Note that this should not to beconfused with the completeness of a space.

Definition 3.10. An orthogonal set {ϕk : k ∈ N} in L2 is called complete iffor any function f ∈ L2 we have

n∑k=1


ϕkL2

→ f.

This actually implies that a complete orthogonal set in L2 is a basis for thespace and this complete orthogonal set is an infinite set since L2 is infinite-dimensional.So if Bessel’s inequality given in (26) is an equality we have

‖f‖2 =

∞∑k=1

|〈f, ϕk〉|2

‖ϕk‖2, (27)

which is called Parseval’s identity and we will relate this identity to definition3.10 about completeness in the following theorem.

Theorem 3.11. An orthogonal set {ϕk : k ∈ N} is complete if and only if itsatisfies Parseval’s identity given in (27) for any f ∈ L2.

As we have seen before the orthogonal set {ϕk} can be normalized to the or-

thonormal set{ψk = ϕk

‖ϕk‖

}. Then Bessel’s inequality can be written as

∞∑k=1

|〈f, ψk〉|2 ≤ ‖f‖2, (28)

and Parseval’s identity turns into

‖f‖2 =

∞∑k=1

|〈f, ψk〉|2. (29)

16

3 VECTOR SPACES

Additionally, since for all f ∈ L2 we have ‖f‖ <∞, recall formula (20), Bessel’sinequality implies 〈f, ψn〉 → 0. So it converges in L2, independent of this or-thonormal set {ψk} being complete.

Furthermore, Parseval’s identity can be seen as the Pythagorean theorem forinner product spaces since it is an abstraction from Rn to the more generalsetting of the L2 space. In this case,

∑∞k=1 |〈f, ψk〉|2 is a representation of the

sum of the squares of the projections of a vector on an orthonormal basis which,by Pythagoras, has to equal the squared length of this vector, now representedby ‖f‖2.

17

4 SELF-ADJOINT DIFFERENTIAL OPERATORS

4 Self-Adjoint Differential Operators

We are now able to define self-adjoint differential operators in the L2-space.Therefore we want to find the adjoint of the operator L : L2(I)∩C2(I)→ L2(I)defined as follows

L = p(x)d2

dx2+ q(x)

d

dx+ r(x), (30)

which is exactly the same operator as we have seen in section 2.1, though using amore convenient notation. Again we assume p, q and r are real C2 functions onI. In order to examine the adjoint of the operator L, which will be denoted byL′, we have to look at the definition of the adjoint. It is given by the followingequation

〈Lf, g〉 = 〈f, L′g〉 for all f, g ∈ L2(I) ∩ C2(I). (31)

We will now further investigate 〈Lf, g〉 to shift the differential operator fromf to g. Then, using equation (31), we obtain an expression for the adjointoperator L′. Setting I = (a, b) and using operator (30), inner product (17) andintegration by parts, we have

〈Lf, g〉 =

∫ b

a

(pf ′′ + qf ′ + rf)g dx

= pf ′g|ba −∫ b

a

f ′(pg)′ dx+ qfg|ba −∫ b

a

f(qg)′ dx+

∫ b

a

frg dx

= (pf ′g − f(pg)′) |ba +

∫ b

a

f(pg)′′ dx+ qfg|ba −∫ b

a

f(qg)′ dx+

∫ b

a

frg dx

= 〈f, (pg)′′ − (qg)′ + rg〉+ (p(f ′g − fg′) + (q − p′)fg) |ba,

which is well defined if p ∈ C2(a, b), q ∈ C1(a, b) and r ∈ C(a, b). If we rewritethe inner product on the right-hand side using a new differential operator de-noted by L∗, we obtain

〈Lf, g〉 = 〈f, L∗g〉+ (p(f ′g − fg′) + (q − p′)fg) |ba. (32)

Rewriting the inner product this way implies the operator L∗ has to satisfy thefollowing equality

L∗g = (pg)′′ − (qg)′ + rg

= pg′′ + (2p′ − q)g′ + (p′′ − q′ + r)g.

Therefore this operator is defined as follows

L∗ = pd2

dx2+ (2p′ − q) d

dx+ (p′′ − q′ + r), (33)

and we will refer to it as the formal adjoint of L. L is formally self-adjoint ifL∗ = L. So comparing equations (33) and (30) yields

p = p, 2p′ − q = q and p′′ − q′ + r = r.

18


This implies q = p′ since we assumed p and q are real. Substituting this in thedefinition of our operator L gives

Lf = pf ′′ + p′f ′ + rf = (pf ′)′ + rf,

which enables us to write the operator L in the following form if it is formallyself-adjoint

L =d

dx

(p

d

dx

)+ r.

Since L = L∗ and q = p′ when L is formally self-adjoint, we can rewrite equation(32) to

〈Lf, g〉 = 〈f, Lg〉+ p(f ′g − fg′)|ba. (34)

Now we are able to determine the form of the self-adjoint operator. We havealready defined the adjoint of L in equation (31) and the operator L is calledself-adjoint if in addition L′ = L. Looking at equation (34) we see this requiresthe following condition

p(f ′g − fg′)|ba = 0 for all f, g ∈ L2(I) ∩ C2(I). (35)

So we have seen the conditions for an operator to be self-adjoint and we willnow apply this to the Sturm-Liouville eigenvalue problem. This is a systemconsisting of the following equation

Lu+ λu = 0, (36)

combined with certain boundary conditions. This equation will be investigatedto find the the eigenvalues λ and their corresponding eigenfunctions for the op-erator −L. Since L turns out to have negative eigenvalues for positive p, we lookfor the eigenvalues of −L. This is possible since −L is (formally) self-adjoint ifand only if the operator L itself is (formally) self-adjoint.

Thus far we have obtained some conditions for the self-adjoint operator in thissection and they will be summarized in the following theorem.

Theorem 4.1. Let L : L2(a, b) ∩ C2(a, b) → L2(a, b) be a second-order lineardifferential operator defined as follows

Lu = p(x)u′′ + q(x)u′ + r(x)u, x ∈ (a, b),

where p ∈ C2(a, b), q ∈ C1(a, b) and r ∈ C(a, b). Then the following statementshold

i) L is formally self-adjoint, i.e. L∗ = L, if the coefficients p, q and r arereal and q = p′.

ii) L is self-adjoint, i.e. L′ = L, if L is formally self-adjoint and equation(35) holds.

19


iii) If L is self-adjoint this implies the eigenvalues λ of equation (36) are allreal and when two eigenvalues are distinct, their corresponding eigenfunc-tions are orthogonal in L2(a, b).

Proof. The first two statements have already been discussed and demonstratedin this section, so we’re left with proving the last statement. Let’s assume λ ∈ Cis an eigenvalue of −L. This implies there exists an associated eigenfunctionf ∈ L2(a, b) ∩ C2(a, b) which is nonzero and for which the following equationholds

Lf + λf = 0.

Therefore we must have

λ‖f‖2 = 〈λf, f〉 = −〈Lf, f〉, (37)

where we substituted λf = −Lf . Since we assumed L is self-adjoint we alsohave

−〈Lf, f〉 = −〈f, Lf〉 = 〈f, λf〉 = λ‖f‖2. (38)

Combining equations (37) and (38) gives

λ‖f‖2 = λ‖f‖2,

which implies λ = λ since ‖f‖ 6= 0. Since λ was chosen arbitrarily, all eigenvalueshave to be real.Suppose we have another eigenvalue µ of −L with corresponding eigenfunctiong ∈ L2(a, b) ∩ C2(a, b). Then we have

λ〈f, g〉 = −〈Lf, g〉 = −〈f, Lg〉 = µ〈f, g〉,

which implies (λ−µ)〈f, g〉 = 0. Since we assumed λ 6= µ we must have 〈f, g〉 = 0,so the eigenfunctions f and g are orthogonal. This completes the proof.

The third statement of theorem 4.1 can be generalized to differential operatorswhich are not even formally self-adjoint. This can be done by defining a positivefunction w such that the operator wL is formally self-adjoint. This w can beregarded as a sort of weight function. Multiplying the eigenvalue equation (36)by w yields

wLu+ λwu = 0.

Now we want L := wL to be formally self-adjoint. Therefore we multiply theoperator L given in (30) by w which gives

L = wpd2

dx2+ wq

d

dx+ wr.

We have seen that an operator is formally self-adjoint if the second coefficientequals the derivative of the first coefficient. For this operator L that is

wq = (wp)′ = w′p+ wp′,

20


and its solution is given by

w(x) =c

p(x)exp

(∫ x

a

q(t)

p(t)dt

),

where c is a constant. We see when L itself is formally self-adjoint, i.e. q = p′,w(x) becomes a constant as we expected. Furthermore if the equivalent of (35),given by

wp(f ′g − fg′)|ba = 0

still holds, the operator wL is even self-adjoint. Now it’s not hard to show thatfor positive function w which is continuous on [a, b] the following statementsagain hold

i) every eigenvalue of Lu+ λwu = 0 is real and

ii) if two eigenfunctions are distinct, they are orthogonal.

This is a generalisation of the third statement in theorem 4.1 since it is notrequired for the operator L to be (formally) self-adjoint anymore.

21

5 THE SPECTRAL THEORY

5 The Spectral Theory

In this section we will discuss the Sturm-Liouville problem and the spectralproperties related to the Sturm-Liouville differential operator. Again we con-sider the following formally self-adjoint operator

L =d

dx

(p(x)

d

dx

)+ r(x), (39)

with the following eigenvalue equation

Lu+ λw(x)u = 0.

If it satisfies the following separated homogeneous boundary conditions

α1u(a) + α2u′(a) = 0, |α1|+ |α2| > 0,

β1u(b) + β2u′(b) = 0, |β1| + |β2| > 0,

(40)

with αi, βi ∈ R, we call this the Sturm-Liouville eigenvalue problem. Fromthe last section we know the function w makes L self-adjoint and thereforethe eigenvalues are real and their corresponding eigenfunctions are orthogonal.Furthermore we note that 0 is not an eigenvalue of this operator. The proof iselementary and will therefore be omitted. As stated in section 2.1 the problemis called regular if the interval (a, b) is bounded and p 6= 0 on [a, b], otherwiseit’s called singular. We will actually only consider the regular problem and inthis case we are able to assume without loss of generality that p(x) > 0. Thenthe eigenfunctions of the operator −L

w solve the problem.

The goal of this section is to prove that the solutions of the Sturm-Liouvilleproblem span the whole space L2(a, b). Throughout this section we shall takew(x) = 1 since it simplifies proofs and calculations, but still covers the generalidea. Eventually we will analyse the spectral properties of L−1 which is relatedto our original operator in the following way. The eigenfunctions of −L coincidewith the eigenfunctions of −L−1 and its eigenvalue equation is given by

L−1u+ µu = 0.

In this case the eigenvalues are related by the identity µ = 1λ . This analysis of

the spectral properties will be done by exploring an integral expression for L−1

denoted by the operator T , making use of Green’s function. This is a continuousC2 function G : [a, b] × [a, b] → R which is symmetric for the Sturm-Liouvilleeigenvalue problem and satisfies the following equation

LxG(x, ξ) = p(x)Gxx(x, ξ) + p′(x)Gx(x, ξ) + r(x)G(x, ξ) = 0, (41)

whenever x 6= ξ. Furthermore its derivative with respect to x has a jumpdiscontinuity at ξ which is given by

∂G

∂x

(ξ+, ξ

)− ∂G

∂x

(ξ−, ξ

)=

1

p (ξ). (42)

22


5.1 Existence of the eigenvalues and eigenfunctions

First the existence of the eigenvalues and the eigenfunctions of the Sturm-Liovuille problem will be proven. Therefore we will build up this Green’s func-tion for the operator L satisfying the boundary conditions given in (40). Usingthe existence and uniqueness theorem for second-order differential equations,we know there must exist two solutions v1 and v2 to the eigenvalue equationLu = 0 which are unique and nontrivial. Furthermore they satisfy

v1(a) = α2, v′1(a) = −α1,

v2(b) = β2, v′2(b) = −β1.

Using these values for v1 and v2 we see they satisfy the boundary conditionsgiven in (40) since we have

α1v1(a) + α2v′1(a) = α1α2 − α2α1 = 0,

β1v2(b) + β2v′2(b) = β1β2 − β2β1 = 0.

Since 0 is not an eigenvalue of the operator L, the solutions v1 and v2 must belinearly independent. Now the Green’s function is defined in the following way

G(x, ξ) =

{c−1v1(ξ)v2(x), a ≤ ξ ≤ x ≤ b,c−1v1(x)v2(ξ), a ≤ x ≤ ξ ≤ b, (43)

where c = p(x) (v1(x)v′2(x)− v′1(x)v2(x)) . Since p 6= 0, proving this c is anonzero constant boils down to showing the derivative of p(x)(v1(x)v′2(x) −v′1(x)v2(x)) equals zero as follows.

Proof.

0 = v1Lv2 − v2Lv1 = v1

((pv′2)

′+ rv2

)− v2

((pv′1)

′+ rv1

)= v1 (pv′2)

′+ v1rv2 − v2 (pv′1)

′ − v2rv1 = v1 (pv′2)′ − v2 (pv′1)

′

= v1p′v′2 + v1pv

′′2 − v2p′v′1 − v2pv′′1

= p′v1v′2 − p′v′1v2 + pv1v

′′2 + pv′1v

′2 − pv′1v′2 − pv′′1 v2

= (p (v1v′2 − v′1v2))

′,

which is known as the Langrange identity.

As noted before, the Green’s function defined in (43) is indeed symmetric andactually satisfies equations (41) and (42). Finally, using G we are able to definethis operator T which we wanted to be an integral expression for L−1. Wewill prove this by showing that the function Tf solves the differential equationLu = f and also the reverse, i.e. Lu solves Tf = u. Therefore we must showTf is a C2 function as well, but let us first define this operator T as follows

(Tf) (x) =

∫ b

a

G(x, ξ)f(ξ) dξ. (44)

Using the continuity of G and f at ξ = x, the continuity of v1 and v2 andthe property of G given in (42) we obtain the following three expressions by

23


differentiating equation (44)

(Tf)(x) =

∫ x

a

G(x, ξ)f(ξ) dξ +

∫ b

x

G(x, ξ)f(ξ) dξ,

(Tf)′(x) =

∫ x

a

Gx(x, ξ)f(ξ) dξ +

∫ b

x

Gx(x, ξ)f(ξ) dξ,

(Tf)′′(x) =

∫ x

a

Gxx(x, ξ)f(ξ) dξ +

∫ b

x

Gxx(x, ξ)f(ξ) dξ +f(x)

p(x).

The elaborations of the first and second derivative can be found on page 13 of[1]. Because of the jump discontinuity of the Green’s function at x = ξ given in(42) these proofs involve some subtleties wherefore they will be left out of thisthesis. Moreover, in the second expression for the first derivative (Tf)′(x) wemade use of the continuity of G and f at ξ = x and we note that the derivativesof limits cancel out.From the last expression it follows that Tf ∈ C2([a, b]). Using the formulasabove and the equation for LxG(x, ξ) given in (41), applying the operator L onthe function Tf yields

L(Tf)(x) = p(x)(Tf)′′(x) + p′(x)(Tf)′(x) + r(x)(Tf)(x)

=

∫ x

a

LxG(x, ξ)f(ξ) dξ +

∫ b

x

LxG(x, ξ)f(ξ) dξ + f(x)

= f(x),

as desired. We also made use of the property of the Green’s function given byequation (41) from which we know LxG(x, ξ) = 0 for every ξ 6= x. Furthermore,because G is symmetric it follows that Tf satisfies the boundary conditions ofthe Sturm-Liouville eigenvalue problem given in (40).On the other hand, if we have another function u ∈ C2([a, b]) which satisfies thesame boundary conditions, we can prove the reverse statement. Making use ofthe fact that p, u and u′ are continuous and also using the properties which theGreen’s function satisfies, we obtain

T (Lu)(x) = u(x).

This is just a matter of substituting Lu(x) in equation (44) and applying inte-gration by parts to the result. Therefore the proof will be omitted.So we have actually seen L(Tf) = f and T (Lu) = u, thus the operator T can beconsidered as an inverse of L. This is exactly the operator we searched for sinceit is an integral expression of L−1. So we see the Sturm-Liouville eigenvalueequation given by

Lu+ λu = 0,

subject to the separated homogeneous boundary conditions given in (40) corre-sponds to the eigenvalue equation

Tu = µu,

with µ = − 1λ . Therefore the eigenfunction u is a solution of the Sturm-Liouville

problem corresponding to the the eigenvalue λ if and only if u is an eigenfunc-tion of T corresponding to the eigenvalue − 1

λ . Since it takes significantly less

24


effort, we shall explore the spectral properties of the integral operator T , asthey present information about the spectral properties of the Sturm-Liouvilleproblem itself.

Let us first look at the eigenvalues of the Sturm-Liouville problem. We havealready noted 0 is not an eigenvalue of −L and the next lemma proves thereexist more real numbers which are not an eigenvalue of this operator.

Lemma 5.1. The eigenvalues of −L are bounded below by a real constant.

Proof. Consider a function u ∈ C2([a, b]) satisfying the boundary conditionsgiven in (40). To say something about the value of the eigenvalues we will firstevaluate the inner product of −Lu with u using (17). Applying (39) and usingthe boundary conditions to replace u′ we get

〈−Lu, u〉 = 〈−(pu′)′ − ru, u〉 =

∫ b

a

(−(pu′)′u− r|u|2

)dx

= [−p(x)u′(x)u(x)]ba −

∫ b

a

−p|u′|2dx+

∫ b

a

−r|u|2dx

=

∫ b

a

(p|u′|2 − r|u|2

)dx+ p(a)u′(a)u(a)− p(b)u′(b)u(b)

=

∫ b

a

(p|u′|2 − r|u|2

)dx+ p(b)

β1β2u2(b)− p(a)

α1

α2u2(a),

where we made use of integration by parts in the second step. If α2 or β2 equals0 the boundary conditions imply u(a) = 0 and u(b) = 0 respectively. Thereforethe second or third term above drops out. In case they both equal zero we canalready state the following concerning the eigenvalue λ corresponding to theeigenfunction u

λ‖u‖2 = 〈−Lu, u〉 =

∫ b

a

p(x)|u′(x)|2 dx−∫ b

a

r(x)|u(x)|2 dx

≥ −‖u‖2max{|r(x)| : a ≤ x ≤ b}.

So if we define ` := −max{|r(x)| : a ≤ x ≤ b}, we see this ` is a lower bound forλ since u ∈ C2([a, b]) was chosen arbitrary.On the other hand if α2 and β2 do not equal zero, we can show by contradiction−L does not have more than two linearly independent eigenfunctions less thanthis lower bound `. So let’s assume our operator has three linearly independenteigenfunctions u1, u2 and u3. Corresponding to them we have the eigenvaluesλ1, λ2 and λ3 respectively, which all three are less than `. Furthermore wesuppose the eigenfunctions are orthonormal. As a reminder, this implies theinner product of ui with uj for i 6= j equals zero and the norm of every uiequals 1. Now we define the following eigenfunction

v(x) = c1u1(x) + c2u2(x) + c3u3(x),

where c1, c2 and c3 are constants and for which we have v(a) = v(b) = 0.This follows directly from the fact that u1, u2 and u3 must satisfy the boundary

25


conditions given in (40), which implies the following two identities hold

c1u1(a) + c2u2(a) + c3u3(a) = 0,

c1u1(b) + c2u2(b) + c3u3(b) = 0.

Since v(a) = v(b) = 0 we must have from the argument above that the eigenvaluecorresponding to v is bounded below by `. However, here we stumble upon acontradiction since the following inequality must also hold

〈−Lv, v〉 = λ1|c1|2 + λ2|c2|2 + λ3|c3|2 < `(|c1|2 + |c2|2 + |c3|2

)= `‖v‖2,

where we used the orthonormality property of the eigenfunctions ui. Thereforethere exist at most two linearly independent eigenvalues less than ` in the caseof α2 and β2 being unequal to zero and none less than ` if α2 and β2 both equalzero. So the eigenvalues of −L are bounded below by a real constant.

By determining an eigenvalue of T we are able to show its existence. Thereforewe first want to prove the set of functions Tu contains a sequence which is uni-formly convergent on [a, b] to a continuous function. We shall see this functionis an eigenfunction of the operator T corresponding to the eigenvalue we aresearching for. To prove that it contains a uniformly convergent sequence we willuse the following theorem, also known as the Ascoli-Arzela theorem which canbe found on pages 28-31 of [9].

Theorem 5.2. Let F be an infinite, uniformly bounded, and equicontinuousset of functions on the bounded interval [a, b]. Then F contains a sequence(fk : k ∈ N) which is uniformly convergent on [a, b] to a function which iscontinuous on [a, b].

Before proving {Tu} is uniformly bounded and equicontinuous we first want toclarify these concepts. Again consider the infinite set F consisting of functionswhich are continuous on [a, b] satisfying the following inequality

|f(x)| ≤M for all f ∈ F and allx ∈ [a, b],

where M is a positive number. Then we call the set F uniformly bounded. Onthe other hand, if for every ε > 0 there exists a δ > 0 such that |f(x)−f(ξ)| < εfor all f ∈ F and all x, ξ ∈ [a, b] whenever |x− ξ| < δ, then we call the infiniteset F equicontinuous on [a, b]. Note that δ may depend on ε, but not on x, ξ orf . Making use of these concepts, we are now able to prove the next theorem byshowing Tu is uniformly bounded and equicontinuous, and subsequently usingthe Ascoli-Arzela theorem.

Theorem 5.3. The set of functions Tu, with u ∈ C([a, b]) and ‖u‖ ≤ 1, con-tains a sequence which is uniformly convergent to a continuous function on[a, b].

Proof. First of all we know |G(x, ξ)| is uniformly continuous and bounded bysome positive constant M since the Green’s function G is continuous on [a, b]×[a, b]. Therefore, consecutively using the expression for the operator T given in(44), the inner product given in (17) and the Cauchy-Schwarz inequality (19),

26


we obtain

|Tu(x)| =

∣∣∣∣∣∫ b

a

G(x, ξ)u(ξ) dξ

∣∣∣∣∣ =

∣∣∣∣∣∫ b

a

G(x, ξ)u(ξ) dξ

∣∣∣∣∣= |〈G(x, ξ), u(ξ)〉| ≤ ‖G‖‖u‖ ≤M

√b− a‖u‖.

(45)

Since we assumed ‖u‖ ≤ 1 we see the set Tu is uniformly bounded since|Tu(x)| ≤ M

√b− a‖u‖ ≤ M

√b− a. On the other hand, again using the uni-

form continuity of G we must have that for any ε > 0 there exists a δ > 0 suchthat for x1, x2 ∈ [a, b], |x2 − x1| < δ implies |G(x2, ξ) − G(x1, ξ)| < ε for allξ ∈ [a, b]. Since u ∈ C([a, b]) we therefore have that whenever |x2− x1| < δ thisimplies |Tu(x2)−Tu(x1)| ≤ ε

√b− a‖u‖ ≤ ε

√b− a. So besides being uniformly

bounded we notice the set of functions Tu for which we have u ∈ C([a, b]) and‖u‖ ≤ 1, is also equicontinuous since our choice for δ is independent of x1, x2and u.

Applying the Ascoli-Arzela theorem given in theorem 5.2 we see we have provenTu contains a uniformly convergent subsequence and this enables us to makea claim concerning the existence of an eigenvalue of T . Therefore we want todefine the norm of this operator as follows

‖T‖ = sup{‖Tu‖ : u ∈ C([a, b]), ‖u‖ = 1}.

Using this norm it is not too hard to show the first of the following two identitieshold for all continuous functions u on [a, b]

1. ‖Tu‖ ≤ ‖T‖‖u‖ ,(46)

2. ‖T‖ = sup‖u‖=1

|〈Tu, u〉| .

The second statement is not completely trivial but can be found on pages 234-235 in [4]. These identities will be useful in proving the following theorem aboutthe existence of an eigenvalue of T .

Theorem 5.4. Either ‖T‖ or −‖T‖ is an eigenvalue of the operator T .

Proof. First of all, since 〈Tu, u〉 is a real number we have using the secondidentity in (46) that either ‖T‖ = sup〈Tu, u〉 or ‖T‖ = − inf〈Tu, u〉 with thenorm of u being equal to 1. We will assume ‖T‖ = sup〈Tu, u〉 since the prooffor −‖T‖ = inf〈Tu, u〉 is similar. This assumption implies there must exist asequence of continuous functions {uk} on [a, b] for which 〈Tuk, uk〉 → ‖T‖ ask →∞. According to theorem 5.2 the sequence {Tuk} contains a subsequence{Tukj} which converges uniformly to a continuous function which we will denoteby ϕ0. We shall see this ϕ0 actually is an eigenfunction of T and its correspond-ing eigenvalue, which we will denote by µ0, indeed equals ‖T‖. So since {Tukj}converges uniformly to ϕ0 we know

supx∈[a,b]

|Tukj (x)− ϕ0(x)| → 0.

27


Therefore, using (46) we have ‖Tukj −ϕ0‖ → 0 which in turn implies ‖Tukj‖ →‖ϕ0‖. Using this and the fact that 〈Tukj , ukj 〉 → ‖T‖ = µ0 we obtain thefollowing identity

‖Tukj − µ0ukj‖2 = ‖Tukj‖2 + ‖µ0ukj‖2 − 2〈Tukj , µ0ukj 〉= ‖Tukj‖2 + |µ0|2‖ukj‖2 − 2µ0〈Tukj , ukj 〉= ‖Tukj‖2 + µ2

0 − 2µ0〈Tukj , ukj 〉→ ‖ϕ0‖2 + µ2

0 − 2µ20 = ‖ϕ0‖2 − µ2

0.

(47)

Since ‖Tukj − µ0ukj‖2 is always greater than or equal to zero on [a, b] wemust have ‖ϕ0‖2 ≥ µ2

0 ≥ 0. Furthermore, using (46) we have ‖Tukj‖2 ≤‖T‖2‖ukj‖2 = ‖T‖2 = µ2

0. This implies (47) boils down to

0 ≤ ‖Tukj − µ0ukj‖2 ≤ µ20 + µ2

0 − 2µ0〈Tukj , ukj 〉 = 2µ20 − 2µ0〈Tukj , ukj 〉.

Since 〈Tukj , ukj 〉 → µ0 the right hand side of the expression above goes to zero.Therefore we obtain ‖Tukj − µ0ukj‖ → 0. Now we will rewrite the norm ofTϕ0 − µ0ϕ0 in the following way using the triangle inequality given in (19)

0 ≤ ‖Tϕ0 − µ0ϕ0‖ = ‖Tϕ0 − T (Tukj ) + T (Tukj )− µ0Tukj + µ0Tukj − µ0ϕ0‖≤ ‖Tϕ0 − T (Tukj )‖+ ‖T (Tukj )− µ0Tukj‖+ ‖µ0Tukj − µ0ϕ0‖.

If we use the just obtained results we see the left hand side tends to zero as jgoes to infinity. Therefore it follows that ‖Tϕ0 − µ0ϕ0‖ must equal zero, whichimplies Tϕ0(x) = µ0ϕ0(x) for all x ∈ [a, b] because Tϕ0 − µ0ϕ0 is continuous.This completes the proof since we have shown that µ0 = ‖T‖ is an eigenvalueof the operator T corresponding to the eigenfunction ϕ0.

We shall now use this result in the proof of the following theorem regarding theexistence of eigenfunctions of T .

Theorem 5.5. The operator T has an infinite sequence of orthonormal eigen-functions in the L2(a, b) space.

Proof. We will prove this theorem inductively, creating a sequence of eigen-functions in the following way. Using ϕ0, µ0, the Green’s function G and theself-adjoint operator T we shall define the normalized eigenfunction ψ0, a newfunction G1 and a new self-adjoint operator T1 as follows

ψ0 :=ϕ0

‖ϕ0‖,

G1(x, ξ) := G(x, ξ)− µ0ψ0(x)ψ0(ξ),

(T1u)(x) :=

∫ b

a

G1(x, ξ)u(ξ) dξ =

∫ b

a

(G(x, ξ)− µ0ψ0(x)ψ0(ξ)

)u(ξ) dξ

=

∫ b

a

G(x, ξ)u(ξ) dξ − µ0ψ0(x)

∫ b

a

u(ξ)ψ0(ξ) dξ

= Tu(x)− µ0〈u, ψ0〉ψ0(x),

(48)

which holds for all continuous functions u on [a, b]. Defining it this way we seeG and G1 satisfy the same regularity properties and moreover G1 is symmetric,

28


i.e. G1(x, ξ) = G1(ξ, x) for all x, ξ ∈ [a, b] following from G being symmetricitself. This implies the set of functions T1u is also uniformly bounded andequicontinuous for ‖u‖ ≤ 1. Furthermore, if we define |µ1| := sup |〈T1u, u〉|for ‖u‖ = 1 and assume ‖T1‖ 6= 0, we see ‖T1‖ = µ1 is an eigenvalue of theself-adjoint operator T1. It corresponds to the continuous eigenfunction ϕ1 andtherefore it satisfies the following equation

T1ϕ1 = µ1ϕ1.

Because ϕ0 is an eigenfunction of T corresponding to the eigenvalue µ0 we mustalso have Tψ0 = µ0ψ0 for the normalized eigenfunction ψ0. Using this we obtain

〈T1u, ψ0〉 = 〈Tu− µ0〈u, ψ0〉ψ0, ψ0〉 = 〈Tu, ψ0〉 − µ0〈〈u, ψ0〉ψ0, ψ0〉= 〈u, Tψ0〉 − µ0〈u, ψ0〉 = 〈u, µ0ψ0〉 − µ0〈u, ψ0〉 = 0,

(49)

where we made use of the operator T being self-adjoint and the eigenfunctionψ0 being normalized, thus its norm being equal to 1. Since this holds for allcontinuous functions u on [a, b] we can now normalize the eigenfunction ϕ1 of T1,i.e. ψ1 := ϕ1

‖ϕ1‖ and see ψ1 is orthogonal to ψ0. This results from substituting

ψ1 for u in expression (49) as follows

0 = 〈T1ψ1, ψ0〉 = 〈µ1ψ1, ψ0〉 = µ1〈ψ1, ψ0〉.

Since we assumed µ1 6= 0 we see the inner product of ψ1 and ψ0 equals zerowhich implies they are orthogonal. Using the definition of T1 given in (48),applying this operator to ψ1 yields

T1ψ1 = Tψ1 − µ0〈ψ1, ψ0〉ψ0 = Tψ1, (50)

where we used the orthogonality of ψ0 with ψ1. Since T1ψ1 = µ1ψ1 we seeTψ1 = T1ψ1 = µ1ψ1 which implies ψ1 is also an eigenfunction of the operator Twith corresponding eigenvalue µ1. Using expression (50), the definitions of µ0

and µ1, and the inequality given in (46) we obtain

|µ1| = |µ1|‖ψ1‖ = ‖µ1ψ1‖ = ‖Tψ1‖ ≤ ‖T‖‖ψ1‖ = ‖T‖ = |µ0|.

In the same way we shall construct G2 and T2 for which we obtain a neweigenfunction. Let’s define them as follows

G2(x, ξ) := G1(x, ξ)− µ1ψ1(x)ψ1(ξ) = G(x, ξ)− µ0ψ0(x)ψ0(ξ)− µ1ψ1(x)ψ1(ξ)

= G(x, ξ)−1∑k=0

µkψk(x)ψk(ξ),

(T2u)(x) :=

∫ b

a

G2(x, ξ)u(ξ) dξ =

∫ b

a

(G(x, ξ)−

1∑k=0

µkψk(x)ψk(ξ)

)u(ξ) dξ

=

∫ b

a

G(x, ξ)u(ξ) dξ −1∑k=0

(µkψk(x)

∫ b

a

u(ξ)ψk(ξ) dξ

)

= Tu(x)−1∑k=0

µk〈u, ψk〉ψk(x).

29


From this we obtain another normalized eigenfunction of T which we will de-note by ψ2 corresponding to the eigenvalue µ2. Again we are able to showthis eigenfunction is orthonormal to the other two eigenfunctions ψ0 and ψ1

and moreover the eigenvalue satisfies |µ2| ≤ |µ1|. Proceeding this way wecan construct a sequence of eigenfunctions ψ0, ψ1, ψ2, . . . of the operator Twhich are orthonormal, and they are associated with the sequence of eigen-values |µ0| ≥ |µ1| ≥ |µ2| ≥ . . . . Then we obtain the following expressions forthe iterated Green’s function and integral operator

Gn(x, ξ) = G(x, ξ)−n−1∑k=0

µkψk(x)ψk(ξ),

(Tnu)(x) =

∫ b

a

Gn(x, ξ)u(ξ) dξ = Tu(x)−n−1∑k=0

µk〈u, ψk〉ψk(x),

(51)

where the norm of the operator Tn satisfies

‖Tn‖ = |µn|. (52)

Obviously, if we have |µn| = ‖Tn‖ = 0 for some n, this sequence of eigenvaluesends. However, it is not too hard to prove that ‖Tn‖ is greater than zero for alln ∈ N. This can be shown by a contradiction, so assuming |µn| = 0 we have

0 = L(0) = L(µnu) = L(Tnu) = L(Tu)− L

(n−1∑k=0

µk〈u, ψk〉ψk

)

= u−n−1∑k=0

µk〈u, ψk〉Lψk ⇒ u =

n−1∑k=0

µk〈u, ψk〉Lψk

=

n−1∑k=0

〈u, ψk〉Lµkψk =

n−1∑k=0

〈u, ψk〉LTψk =

n−1∑k=0

〈u, ψk〉ψk.

But this must hold for all continuous functions u on [a, b] which contradictsthe fact that there cannot exist a finite set of functions which spans wholeC([a, b]). Therefore there must exist an infinite sequence of eigenfunctions,which completes the proof.

5.2 Completeness of the eigenfunctions

In this section we shall prove that the eigenfunctions of the operator T arecomplete. Therefore we make use of theorem 3.11 in which Parseval’s identityis related to the completeness property of an orthogonal set. Thus we have toprove that Bessel’s inequality given in (28) is an equality, i.e.

f =

∞∑k=0

〈f, ψk〉ψk.

The following theorem will be used to show this holds for any f ∈ L2(a, b).

Theorem 5.6. Given any f ∈ C2([a, b]) subject to the separated homogeneousboundary conditions given in (40) the infinite series

∑〈f, ψk〉ψk is uniformly

convergent to f on [a, b].

30


Proof. First we will prove∑∞k=0 µk〈u, ψk〉ψk is uniformly convergent to a con-

tinuous function on [a, b]. As we constructed the eigenfunctions for the operatorT we saw for any function 〈u, ψk〉ψk we have T (〈u, ψk〉ψk) = µk (〈u, ψk〉ψk).This implies

T

(n∑

k=m

〈u, ψk〉ψk

)=

n∑k=m

µk〈u, ψk〉ψk,

for n > m. From expression (45) we know the set Tu is uniformly bounded since|(Tu)(x)| ≤ M

√b− a‖u‖ for all continuous functions u on [a, b]. Therefore,

using the identity above and the definition of the norm given in (18) we obtain∣∣∣∣∣n∑

k=m

µk〈u, ψk〉ψk

∣∣∣∣∣ =

∣∣∣∣∣T(

n∑k=m

〈u, ψk〉ψk

)∣∣∣∣∣ ≤M√b− a∥∥∥∥∥

n∑k=m

〈u, ψk〉ψk

∥∥∥∥∥= M

√b− a

√√√√ n∑k=m

∫ b

a

|〈u, ψk〉ψk|2 dx

= M√b− a

√√√√ n∑k=m

|〈u, ψk〉|2,

since ψk is normalized. From Bessel’s inequality given in (28) we know the lastexpression must be less than or equal to M

√b− a‖u‖. So if m,n → ∞ this

implies |∑nk=m µk〈u, ψk〉ψk| → 0 and therefore

∑nk=m µk〈u, ψk〉ψk is uniformly

convergent to a continuous function on [a, b]. Now we will show this continuousfunction turns out to be Tu. Therefore we will apply Bessel’s inequality to theGreen’s function. First we notice by definition of the operator T given in (44)that

〈G(x, ξ), ψk〉 =

∫ b

a

G(x, ξ)ψk dξ = Tψk(x) = µkψk(x),

for every x ∈ [a, b]. Using this expression and applying Bessel’s inequality weobtain

n∑k=0

µ2k|ψk(x)|2 =

n∑k=0

|µkψk(x)|2 =

n∑k=0

|〈G,ψk〉|2 ≤ ‖G‖2 =

∫ b

a

|G(x, ξ)|2 dξ,

where the last equality follows from (18). As noticed before we know the Green’sfunction is bounded by some positive constant M . Therefore, using the fact thatψk is normalized and integrating the expression above with respect to x yields

∞∑k=0

µ2k ≤M2(b− a)2,

as n → ∞. This implies we must have that limn→∞ |µn| = 0. Now using theexpression for Tn given in (51), the value of its norm given in (52) and theinequality given in (46) we obtain for any continuous function u on [a, b]

‖Tu−n−1∑k=0

µk〈u, ψk〉ψk‖ = ‖Tnu‖ ≤ ‖Tn‖‖u‖ = |µn|‖u‖.

31


Since we just determined limn→∞ |µn| = 0, the right hand side of the inequalityabove also tends to zero as n→∞. Because Tu is continuous we therefore have

Tu(x) =

∞∑k=0

µk〈u, ψk〉ψk(x), (53)

for all x ∈ [a, b]. As we saw before, if f is continuous function on [a, b] satisfyingthe boundary conditions of the Sturm-Liouville problem given in (40) we candefine another continuous function u as follows

u = Lf and f = Tu,

since T acts as an inverse operator of L. Using this and the fact that T isformally self-adjoint we obtain

µk〈u, ψk〉 = 〈u, µkψk〉 = 〈u, Tψk〉 = 〈Tu, ψk〉 = 〈f, ψk〉,

as µk is the eigenvalue of the operator T corresponding to the normalized eigen-function ψk. Substituting this into equation (53) we finally obtain

f(x) = Tu(x) =

∞∑k=0

µk〈u, ψk〉ψk(x) =

∞∑k=0

〈f, ψk〉ψk(x),

which holds for all x ∈ [a, b]. Therefore∑〈f, ψk〉ψk is uniformly convergent to

f ∈ C2([a, b]) and this completes the proof.

Now using the density of C2([a, b]) in L2(a, b) already given in theorem 3.9 weare able to prove ∥∥∥∥∥f −

n∑k=0

〈f, ψk〉ψk

∥∥∥∥∥→ 0

as n→∞. The proof goes too much into detail for this thesis and will thereforebe omitted, but it can be found on pages 81-83 in [2]. However, we have shownthat

limn→∞

∥∥∥∥∥f −n∑k=0

〈f, ψk〉ψk

∥∥∥∥∥ = 0 =⇒ f =

∞∑k=0

〈f, ψk〉ψk,

which is equivalent to Parseval’s identity given in (29). As proved before wesee this shows that the set of eigenfunctions of the operator T is orthonormaland furthermore complete in L2(a, b). We are now able to summarize thiswhole section. Using theorem 5.5 and the relation between the eigenvalues ofthe operators T and L given by µ = − 1

λ , we see limn→∞ |µn| = 0 implies1|λn| = |µn| → 0. Considering the eigenvalues are bounded below (lemma 5.1)

we therefore have λn → ∞ as n → ∞. Also taking the weight function w intoaccount we obtain the following fundamental theorem regarding the Sturm-Liouville eigenvalue problem, which was already stated in section 2.2.

Theorem 5.7. Consider the following formally self-adjoint operator

L =d

dx

(p(x)

d

dx

)+ r(x),

32


with the following eigenvalue equation

Lu+ λw(x)u = 0,

satisfying the following separated homogeneous boundary conditions

α1u(a) + α2u′(a) = 0, |α1|+ |α2| > 0,

β1u(b) + β2u′(b) = 0, |β1| + |β2| > 0,

with αi, βi ∈ R. Suppose p′, r, w ∈ C([a, b]) with p, w > 0 and x ∈ [a, b].Then the Sturm-Liouville eigenvalue problem has an infinite sequence of realeigenvalues which can be ordered in the following way

λ0 < λ1 < λ2 < · · · < λn < · · · ,

where λn →∞ as n→∞. The eigenfunctions ϕn corresponding to each eigen-value λn are unique and after normalizing the eigenfunctions to ψn they forman orthonormal basis of the L2(a, b) space.

33

6 APPLICATION OF THE STURM-LIOUVILLE THEORY

6 Application of the Sturm-Liouville Theory

In this section we shall have a look at some applications of the Sturm-Liouvilletheory and an example will be elaborated. This illustrates the main results ob-tained in this thesis. The section ends with some concluding remarks concerningthe Sturm-Liouville problem and its applications.

6.1 Applications

Besides the example of the sound of a guitar mentioned in the introduction,the Sturm-Liouville theory has many more applications. These equations occurfrequently in applied mathematics as well as in physics. They describe the vi-brations of a particular system, e.g. the vibrations of the plucked string of theguitar. Another example is the one-dimensional time-independent Schrodingerwave equation, where the eigenvalues represent the energy levels of the atomicsystem. These Sturm-Liouville problems do not only occur in one space dimen-sion but also in higher dimensions and the results obtained in this thesis thenstill apply. Considering the wave equation, an example for dimension 2 is givenby the resonant frequencies of a drum. If the dimension equals 3 we can thinkof the resonant frequencies of sound waves in a room.One illustration of the one-dimensional Schrodinger equation is given by a crys-tal structure. This wave equation then represents the motion of a conductionelectron in the crystal structure. The spectrum of the Sturm-Liouville eigen-value equation can be seen as existing of intervals. The location of an eigenvaluein one of these intervals determines whether a solution is bounded or unbounded,where in the latter case it grows exponentially. Therefore some electrons canmove freely through the crystal and the motion of others is bounded, since theirenergy lies in either of both intervals. This in turn clarifies whether the crystalacts like an insulator or conducts electricity. An example of a semiconductor isgiven by silicon, which is among other things used to create computer chips.Another example of Sturm-Liouville equations are the Airy functions, which de-scribe the change of a solution from oscillatory to exponential behaviour. Thiscan be clarified by the oscillatory integrals for the Airy functions which havetwo stationary phase points. Illustrations of this transition from one state to theother are the caustics in light reflections or the passage of a particle from oneregion to another in semi-classical quantum mechanics. A very familiar exampleof a caustic is a rainbow.The Airy functions can also be found in the study on linear dispersive waterwaves. For example, Diederik Korteweg and Gustav de Vries came up with amodel consisting of a nonlinear, dispersive partial differential equation whichdescribes waves on shallow water surfaces. Also Kelvin, who actually inventedthe method of stationary phase, studied the pattern of dispersive water wavesmade by a ship with constant speed.A few other Sturm-Liouville equations are Bessel’s equation, Legendre equationsand Laguerre equations. However, they will not be discussed in this thesis buta very interesting lecture on Sturm-Liouvile eigenvalue problems can be foundin chapter 4 of [5].

34


6.2 An example: the second derivative

At last we shall discuss an example which illustrates the concepts treated sofar. We will have a look at one of the simplest forms of a Sturm-Liouvilleproblem, i.e. the second derivative −

(d2/dx2

). In view of equation (30) for

the operator L we see this is the case for p = −1, q = 0 and r = 0. Theorem4.1 now implies this operator is formally self-adjoint since all the coefficients

are real and furthermore p′ = d(−1)dx = 0 = q. To determine its eigenvalues and

eigenfunctions we need to solve the equation Lu = λu, which implies

u′′ + λu = 0. (54)

Let’s assume it’s subject to the following boundary conditions

u(0) = u(l) = 0,

for 0 ≤ x ≤ l. We will explore this for different values of λ, so let’s start with thecase where the eigenvalue is strict positive. From ordinary differential equationswe know the general solution is then given by

u(x) = c1 cos√λx+ c2 sin

√λx,

where c1 and c2 are constants. Using the boundary conditions we obtain

u(0) = c1 = 0,

u(l) = c2 sin√λl = 0.

The last equality implies λn = n2π2

l2 where n ∈ N. Since both the eigenvalueequation and the boundary conditions are homogeneous we can take c2 = 1and obtain the following expression for the eigenfunctions corresponding to theeigenvalues λn

un(x) = sinnπ

lx.

Now considering the case where λ ≤ 0, we obtain the following solutions for thedifferential equation

u(x) = c1x+ c2 if λ = 0,

u(x) = c1 cosh√−λx+ c2 sinh

√−λx if λ < 0.

Using the boundary conditions both cases lead to the trivial solution u = 0 andtherefore we conclude equation (54) only has positive eigenvalues. Thus theeigenvalues and eigenfunctions of this particular Sturm-Liouville problem aregiven by

λn =n2π2

l2and un(x) = sin

nπ

lx for n ∈ N.

So we see the eigenvalues are real, form an infinite sequence and tend to infinityas n→∞, which was also concluded in theorem 5.7. Furthermore we can prove

35


the eigenfunctions are orthogonal in the space L2(0, l) by showing each innerproduct equals zero. For if we have m 6= n and use definition (17) we get

〈um, un〉 =

∫ l

0

sinmπ

lx sin

nπ

lxdx

=1

2

∫ l

0

(cos(m− n)

π

lx− cos(m+ n)

π

lx)

dx = 0,

since m and n are integers. Finally, using theorem 5.7 we know the sequence{un} of eigenfunctions is complete in L2(0, l) and spans this space.The completeness of the eigenfunctions implies we are able to represent anyfunction f ∈ L2(0, l) as follows

f(x) =

∞∑n=1

αn sinnπ

lx. (55)

Note the equality holds in the L2(0, l) space, which means it should be inter-preted as follows ∥∥∥∥∥f(x)−

∞∑n=1

αn sinnπ

lx

∥∥∥∥∥ = 0.

In section 3.2.2 we already found an expression for the coefficient αn given in(23), which for this example turns into

αn =〈f, sin(nπx/l)〉‖ sin(nπx/l)‖2

, (56)

where we substituted the eigenfunction un(x) we just found. Evaluating thedenominator using (18) yields∥∥∥sin

nπ

lx∥∥∥2 =

∫ l

0

sin2 nπ

lx dx =

l

2.

Substituting this in our formula for the coefficient αn given in (56) and evalu-ating the inner product we then obtain

αn =2

l

∫ l

0

f(x) sinnπ

lx dx.

This can be illustrated by showing how to represent the constant function f(x) =1 on [0, l]. The inner product of f(x) with our eigenfunction un is given by

〈1, sin(nπx/l)〉 =

∫ l

0

sinnπ

lxdx =

l

nπ(1− cosnπ) =

l

nπ(1− (−1)n).

Therefore our coefficient αn turns into

αn =2

l〈1, sin(nπx/l)〉 =

2

nπ(1− (−1)n)

36


Substituting this in our formula for the representation of f given in (55) yields

1 =2

π

∞∑n=1

(1− (−1)n

n

)sin

nπ

lx,

where again the equality is in the space L2(0, l). Therefore we should interpretit as

2

π

n∑k=1

(1− (−1)k

k

)sin

kπ

lxL2

→ 1,

when n→∞.

6.3 Concluding remarks

In this section we have discussed some applications of the Sturm-Liouville theoryand we looked at an example. This illustrated some of the spectral propertiessuch as the infinite sequence of real eigenvalues and the representation of a func-tion in the L2 space. We saw the set of functions {sin(nπx/l) : n ∈ N} spansL2(0, l). It can be shown that {cos(nπx/l) : n ∈ N} also spans this space andtheir combination spans L2(−l, l), leading to the Fourier series. However weshall not further digress on this, but it shows how comprehensive the Sturm-Liouville problem is. We could also have considered the singular case, i.e. whensome of the initial conditions are not satisfied. For example the function p(x)being equal to zero at any of the endpoints a or b, or the interval (a, b) beinginfinite. Then the spectral properties given in this paper are still satisfied andit even leads to a generalisation of the theory and its conditions. Well-knownexamples follow from this singular problem, e.g. Legendre’s and Hermite’s equa-tion, or the one-dimensional time-independent Schrodinger equation. Thus wesee the Sturm-Liouville theory is deeply investigated and utilized and still is tothis day. Hence this thesis did not cover the entire research on this topic but itdid work towards some of its main results to make the reader familiar with theSturm-Liouville theory.

37

7 ACKNOWLEDGEMENTS

7 Acknowledgements

I would like to express my gratitude to my first supervisor dr. A.E. Sterk.Knowing my research would take longer, I chose to ask him to supervise mymaster thesis since he also guided me through my bachelor’s project. I want tothank him for his adequate feedback and for his patience to let me work at myown time. He motivated me with his own passion for this topic and I appreciatehis advise and suggestions when I encountered difficulties. It was an honourand pleasure to have him as my supervisor again.

I am also thankful to my family and friends for their support during my educa-tion. I could always rely on them and I appreciate their encouragements.

Groningen, July 2017David Koning

38

8 REFERENCES

8 References

[1] Stochastic Systems. Mathematics in Science and Engineering. Elsevier Sci-ence, 1983.

[2] M. Al-Gwaiz. Sturm-Liouville Theory and its Applications. Springer Un-dergraduate Mathematics Series. Springer London, 2008.

[3] N.L. Carothers. Real Analysis. Cambridge University Press, 2000.

[4] M. Haase. Functional Analysis: An Elementary Introduction. GraduateStudies in Mathematics. American Mathematical Society, 2014.

[5] John K. Hunter. Lecture Notes On Applied Mathematics. University ofCalifornia, 2009.

[6] S.J. Leon. Linear Algebra with Applications: Pearson New InternationalEdition. Pearson Education, Limited, 2013.

[7] J. Lutzen. Joseph Liouville, 1809-1882, master of pure and applied math-ematics. Studies in the history of mathematics and physical sciences.Springer-Verlag, 1990.

[8] Y. Pinchover and J. Rubinstein. An Introduction to Partial DifferentialEquations. Number v. 10 in An introduction to partial differential equa-tions. Cambridge University Press, 2005.

[9] M. Reed and B. Simon. I: Functional Analysis. Methods of Modern Math-ematical Physics. Elsevier Science, 1981.

[10] W. Rudin. Principles of Mathematical Analysis. International series inpure and applied mathematics. McGraw-Hill, 1976.

[11] W.A. Sutherland. Introduction to Metric and Topological Spaces. OUPOxford, 2009.

[12] R. Thompson and W. Walter. Ordinary Differential Equations. GraduateTexts in Mathematics. Springer New York, 2013.

[13] A. Zettl. Sturm-Liouville Theory. Mathematical Surveys and Monographs.American Mathematical Society, 2005.

39

the sturm -liouville problem - university of groningen...abstract this thesis treats the...

Documents